Master Dependent and Independent Variables in CBSE Class 11 Applied Mathematics. Learn to identify variables in functions with clear examples and exercises.
In a mathematical function y = f(x), the variable x is called the independent variable because it can be assigned any value from the domain. The variable y is the dependent variable because its value is determined by the value assigned to x. This relationship represents a cause-and-effect link between two quantities.
Step 1: Identify the input variable, which is x.
Step 2: Identify the output variable, which is f(x) or y.
Step 3: Since y depends on x, x is independent and y is dependent.
Answer: Independent: x, Dependent: y
When modeling physical or economic scenarios, the independent variable often represents time, quantity, or cost. The dependent variable represents the result, such as total revenue, distance, or profit. Recognizing these roles is essential for graphing functions correctly.
Step 1: Here q represents the quantity produced.
Step 2: C(q) represents the total cost.
Step 3: As quantity q changes, cost C changes.
Answer: Independent: q, Dependent: C
The set of all possible values for the independent variable is the domain, while the set of all resulting values for the dependent variable is the range. For a function to be well-defined, each independent value must map to exactly one dependent value.
Step 1: x must be ≥ 0 for y to be a real number.
Step 2: x is the independent variable.
Step 3: y is the dependent variable.
Answer: Independent: x, Dependent: y